We introduce a new general class of nonlinear variable order fractional partial differential equations (NVOFPDE). The NVOFPDE contains, as special cases, several partial differential equations, such as the nonlinear variable order (VO) fractional equations usually denoted as Klein-Gordon, diffusion-wave and convection-diffusion-wave. To find the numerical solution of the NVOFPDE, we formulate a novel class of basis functions called generalized shifted Chebyshev polynomials (GSCP) that includes the shifted Chebyshev polynomials as a particular case. The solution of the NVOFPDE is expanded following the GSCP and the corresponding operational matrices of VO fractional derivatives (VO-FD), in the Caputo type, are obtained. An optimization method based on the GSCP and the Lagrange multipliers converts the problem into a system of nonlinear algebraic equations. The convergence analysis is guaranteed through a theorem concerning the GSCP and several numerical examples confirm the precision of the method.

我们介绍了一类新的非线性变量阶分数阶偏微分方程（NVOFPDE）。在特殊情况下，NVOFPDE包含几个偏微分方程，例如通常表示为Klein-Gordon的非线性变量阶（VO）分数方程，扩散波和对流扩散波。为了找到NVOFPDE的数值解，我们制定了一种称为通用移位Chebyshev多项式（GSCP）的新型基函数，其中包括移位Chebyshev多项式作为一种特殊情况。根据GSCP扩展NVOFPDE的解，并获得Caputo类型的VO分数导数（VO-FD）的相应运算矩阵。基于GSCP和Lagrange乘数的优化方法将问题转换为非线性代数方程组。通过关于GSCP的一个定理可以保证收敛性分析，并且几个数值例子证明了该方法的准确性。